I am stuck on this one. I know since $(a_n)$ is Cauchy, then $ \forall \epsilon>0$ $\exists n,m \in \mathbb[N]$ so that whenever $n,m \geq N, |a_n - a_m| < \epsilon$.
So I'm thinking, when $n$ is odd, $|-a_n + a_m| < \epsilon$, and when
$n$ is even, $|a_n - a_m| < \epsilon$.
I'm not sure if I'm on the right track here or not.